Abstract
Euclidean $n$-space $E^n$, $n\geqq 5$, has the following simple DISJOINT DISK PROPERTY: singular $2$-dimensional disks in $E^n$ may be adjusted slightly so as to be disjoint. We show that for a large class of cell-like decompositions of manifolds this property in the decomposition space is insufficient in order that the decomposition space be a manifold. As a consequence we deduce the DOUBLE SUSPENSION THEOREM proved in a large number of cases by R. D. Edwards: The double suspension of any homology sphere is a topological sphere. We also obtain a sweeping generalization of Edwards’ MANIFOLD FACTOR THEOREM: Edwards’theorem states that, if $X$ is a single cell-like set in Euclidean $n$-dimensional space $E^n$, then $(E^n/X) \times E^1 = E^{n+1}$.
This paper is dedicated to R. H. Bing and R. D. Edwards whose beautiful ideas made it possible.
